The foundations of statistical inference are very hard to teach at any level, and almost certainly, at the 7th grade level, little or no serious motivation is given for the rules presentd. Probably at the 7th grade level what is being taught are rote rules for calculation of defined, but generally poorly if at all motivated, quantities. The presence or absence of a square is hardly of much importance if that is the situation.
In quantifying error, the power used refects some subjective judgment about the importance attached to large values or outliers. One can use the absolute value, the square, or any $p$th power (or the $\infty$-norm). Different choices are affected differently by large values - the mean error calculated using a higher power is more affected by large values than is the error calculated using a lower power. The widespread use of the square has sound justifications in scientific practice, but the ordinary mean also can make sense. When working with large data sets on a computer, computational considerations can dictate using the square. Fundamentally the preference for the square is because the squared mean is given by an inner product. For example, the regression line defined using mean squared error is more easily calculated on a computer than that defined using mean absolute error because the "normal equations" can be solved efficiently and robustly using the singular value decomposition.
On the other hand, if one refers to computation by hand, then the calculation without squares requires fewer operations and so is simpler. At the $7$th grade level little or no effort is made to interpret, and to the extent that any is made, the difference between ordinary mean and mean squared is not of great importance.
There is nothing any more "real world" about any of this stuff than there is about the standard exercises given to calculus students. One teaches mathematics to all $7$th graders not because of its real world utility, but because the habits of thought learned by learning mathematics are themselves useful quite generally. One teaches elementary statistical ideas in order to develop the ability to formulate qualitative interpretations of masses of data. What one can hope to teach students is that there is some sort of second order criterion that can distinguish distributions with the same mean. One hopes that a student can distinguish (visually/intuitively) between data uniformly distributed over a range and data all repeated on the midpoint of that same range, and then see that the measure of deviation (with or without squares) differentiates between these two data points. This is in the same spirit as teaching the student that distributions with the same mean can have different medians, but it is different, because mean and median are both first order statistics, whereas deviation is somehow second order, measuring spread around something first order. There are different ways to measure spread, but it becomes difficult to justify, on principled grounds understandable at the middle school level, the choice of one measure of spread over another. If I were teaching this material, I would probably use the mean squared error as the default, but I have little experience teaching $7$th graders per se, and am prepared to imagine the leaving out the square facilitates the realization of computations with little conceptual loss of a nature relevant at that level.
